Blues won a brilliant game that had thrills, spills, penalty misses and early baths. Anelka and Baines missed their spot-kicks, whilst Turnbull and Drenthe saw red.
The opening stages of the game ebbed and flowed with both teams coming close to opening the scoring - Chelsea's Romelu Lukaku's second minute shot drifted wide before Louis Saha made keeper Ross Turnbull work twice inside the first 10 minutes.
Josh McEachran caused mayhem in the Toffees' box after quarter of an hour, but his shot was blocked by Sylvain Distin before being brought down for a penalty by Johnny Heitinga. But Nicolas Anelka missed the spot kick and gave the home side respite.
Everton responded quickly after that with Russian Diniyar Bilyaletdinov's shot forcing a save from Turnbull, whilst Heitinga tried to make up for his challenge in the box by shooting from distance but his shot sailed effortlessly wide.
Lukaku and Bilyaletdinov exchanged chances before Salomon Kalou opened the scoring after 37 minutes from just outside the area after good work by Ryan Bertrand. His chipped shot was missed completely by Mucha who tried hopelessly to claw it back before it went over the line.
The second-half proved to be even more entertaining than the first with Kalou and Anelka testing a nervy Mucha before his counterpart Turnbull was sent off for bringing down Saha just after the hour.
Andre Villas-Boas responded by bringing on number one goalkeeper Petr Cech - for the impressive Lukaku - just in time for the penalty and what an impact the Chelsea No.1 made.
The Czech international trumped Leighton Baines' effort, and then produced another save to deny Bilyaletdinov from the rebound to keep his side in front.
But this seemed to merely spur on the hosts, who started to take control of proceedings with their one-man advantage and came close to equalising through Royston Drenthe but his shot just went over.
Mucha-d up | Howler from Everton No.2 gave Chelsea the lead at Goodison Park
With seven minutes remaining Saha headed in substitute Seamus Coleman's cross - who had just come on - as the host's pressure finally paid off.
The impressive Drenthe and Denis Stracqualursi both went close to putting the Toffees in front and Coleman hoofed a Florent Malouda chance away at the other end in the dying embers of the 90.
The last chance of normal time came from goalscorer Saha who shot a foot wide through a host of players.
Everton pushed hard in the first-half of extra-time looking for the all in important winning goal without really threatening the goal, whilst Chelsea relied on the pace of substitute Sturridge on the counter.
But the Toffees quickly lost their advantage, as Drenthe's over enthusiastic challenge on Bertrand saw the hosts reduced to ten men, and handed Chelsea the impetus.
Anelka had a chance to make up for his earlier penalty miss but his shot came back off the post after Baines got back to get a sliding touch on it before Sturridge mis-cued at the wrong time following work from Branislav Ivanovic.
But Sturridge went from villain to hero minutes later, tapping in from close range after Mucha could only parry Malouda's long-range drive to see the visitors in the hat for the next round.